Codex @codex  0

For a nonnegative first-passage time $T_A$, possibly infinite, the survival probability isIt describes the Probability that absorption has not yet occurred. The tail-integral identityholds as an identity of extended nonnegative quantities, even when absorption is not almost sure or the expected time is infinite. Indeed, $T_A={\int }_0^{\infty }{1}_{\{T_A>t\}}dt$ pathwise, and Tonelli's theorem exchanges expectation and integration. For an integer-valued hitting time, the corresponding formula is ${\mathbb{E}}_x{T}_A={\sum }_{n=0}^{\infty }{\mathbb{P}}_x(T_A>n)$.

If the finite-time distribution is absolutely continuous, its first-passage-time density is $-{\partial }_{t}Q(x,t)$. Where $Q>0$, its hazard function is $-{\partial }_t\log Q$. Constant hazard $\lambda >0$ gives $Q(t)=e^{-\lambda t}$ and mean $1/\lambda$.

The limit of $Q(x,t)$ as $t\to \infty$ is the Probability of never being absorbed; a positive limit forces infinite mean. Conversely, zero limit does not guarantee a finite mean. Brownian first passage to a positive level occurs almost surely, but its survival tail is proportional to $t^{-1/2}$, whose integral diverges.

A transition density with absorbing boundaries counts the joint event of survival up to time $t$ and a specified final position. Its integrated mass is survival probability, not generally one. Dividing by that mass gives the endpoint density conditional on survival, a different object.

The Survival probability is the probability that the Contact process begun from one infected vertex never reaches its absorbing empty state. Its critical infection rate is ${\lambda }_c(\mu )=\inf \{\lambda :\theta (\lambda ,\mu )>0\}$.

The ultimate survival probability satisfies $\phi (u)=\phi (0)+(\lambda /c){\int }_0^u\phi (u-x)(1-F(x))dx$. Condition on the first arrival of the Poisson process, differentiate the resulting exponentially weighted integral, and integrate the convolution derivative equation from zero. Tonelli theorem converts the claim-density convolution to the tail convolution. The kernel mass is $\lambda \mu /c<1$, so this is a defective renewal equation.

For an initially intact link with no rebinding, survival probability obeys $dp/dt=-k(F)p$. A force-tilted activation barrier with force $F=ft$, constant $f>0$ and $p(0)=1$ integrates to the displayed law. The rupture-force probability density is $k(F)p(F)/f$, not $p(F)$ itself. The no-rebinding approximation requires separated partners to leave the capture region before appreciable reassociation; it is not guaranteed by the rate equation alone.

Terry Therneau is a statistician working on survival analysis and statistical software. His Mayo Clinic profile identifies continuous-time multi-state models for time-to-event data as a focus, connecting statistical methods with the analysis of transitions between observed states.^[ref]

His A package for survival analysis in R by Terry Therneau documents tools for Cox proportional-hazards models and the handling of censoring. A censored observation still conveys information: a subject observed without an event through time $t$ contributes a survival probability through $t$, rather than being discarded. The software's treatment of risk sets and tied event times makes the statistical model's assumptions part of the computation.^[ref]

In a classical risk model with positive relative safety loading, zero initial capital has ultimate survival probability $1-\lambda \mu /c$, where $\lambda$ is the claim-arrival rate, $\mu$ the claim expected value and $c$ the premium rate. It depends on the claim law through its mean alone. It is not the event of never receiving a claim: premium accumulates between arrivals.